Transcription
The theme of recent talks has been looking at how x-rays interact with matter as they pass through the tissues of our patient, as well as how those interactions result in dose being applied to our patients.
Now, we're going to look at how x-rays are attenuated as they travel through a set distance of tissue. We're going to see how we can calculate the removal of x-rays as it passes through tissues, or at least think about how x-rays are removed or attenuated from an x-ray beam as they go over a set distance of tissue.
Now, the way we do this is by looking at the linear attenuation coefficient of a specific x-ray beam passing through a specific tissue. Now, the linear attenuation coefficient determines the amount of x-rays that are removed from our x-ray beam as it's traveling across a set distance.
Now, we know x-rays can either be transmitted through the patient, attenuated by the photoelectric effect, or scattered via either Compton scatter or Rayleigh scatter. Now, we're looking at the proportion of x-rays that are either attenuated or scattered as opposed to those that are transmitted.
Now, when we're thinking about the linear attenuation coefficient, we think of it as x-rays coming in as incident x-rays. These x-rays are coming from our x-ray machine. They are traveling through a set distance of tissue, represented by axial. This green here is our patient. The width here is the distance those x-rays have to travel. X-rays coming out to the other side of those patients are our transmitted x-rays, those that have not interacted with any of the tissue here.
Now, what I find useful when thinking about the linear attenuation coefficient is reminding myself about what a coefficient is and what the function of a coefficient is. Now, when we look at an equation like this, this is the equation for a straight line graph. Here is a straight line graph. Here, there are multiple factors that make up an equation. We've got variables that you'll know are Y and our X. Our variables determine the axes on our graph. As X changes, Y will change in relation to X. Then we've got constants. Now, constants are exactly what they say they are. They are constant. They don't change, no matter what happens to the variables. The constant determines where our graph will lie in relation to our X and Y axes. Then we have what's known as coefficients. Now, coefficients are linked to a variable, and they serve multiple functions. In a straight line graph, our coefficient determines the gradient of this graph. It determines the rate of change of the graph. A coefficient links our two variables to one another. As X changes, Y will change in proportion to that coefficient. Our coefficient determines the shape of a graph, the rate of change of that graph.
Now, when we're thinking about the linear attenuation coefficient, that's how we should think about it. Our linear attenuation coefficient determines the rate of x-ray removal, determines the shape of that graph, how many x-rays we are removing per distance that those x-rays are traveling. It's a coefficient.
Now, the linear attenuation coefficient, we use the symbol mu here. This is our LAC, our linear attenuation coefficient, and as I've mentioned, it's the combination of both the photoelectric, Compton, and Rayleigh scatter interactions. Those combined make up our linear attenuation coefficient.
Now, when we're thinking about these interactions, photoelectric effect, Compton scatter, and Rayleigh scatter, we know there are certain variables that will determine the probability of those interactions happening. We know that as our x-ray energy increases, the likelihood of those interactions decrease. X-ray attenuation per unit distance decreases. Our linear attenuation coefficient decreases as x-ray energy increases. We know that the tissue's density determines the likelihood of these interactions to occur, as well as the tissue's atomic number. So, it's x-ray energy, tissue density, and tissue atomic number that will determine the rate of these interactions happening, our photoelectric effect, Compton scatter, and Rayleigh scatter. So, it goes without saying that these three variables will determine our linear attenuation coefficient. They will determine the rate or the proportion of x-rays that are removed from a beam over a set distance in our tissues. So, this linear attenuation coefficient will change if any of these three variables change.
So, how do we go about calculating our linear attenuation coefficient? Well, we can look at our incident x-rays and our transmitted x-rays. And I've said our linear attenuation coefficient is the proportion of x-rays that are removed by a set distance in tissue. So, what is that proportion of x-rays that are removed? Well, it's our incident x-rays here, minus the number of transmitted x-rays. That will give us the number of x-rays that have been removed from the beam. Now, our linear attenuation coefficient is the proportion of x-rays that are removed from a beam. So, logically, the number of incident x-rays times by that proportion, the proportion that are removed, should give us the number of x-rays that are removed from a beam. So, if we were to look at this equation, we would reduce that. If the first millimeter of tissue removed 10 x-ray photons, the next millimeter of tissue would remove 10 x-ray photons, and that's not the case. This relationship is not linear. It's actually a negative exponential relationship.
So, this is the equation we actually need to look at when looking at attenuation in a tissue. And this graph here shows the attenuation of x-rays as it travels through a set distance of tissue. Here, we can see that initially, we get a large reduction in the number of x-ray photons within LBM. And as we travel through that tissue, the absolute number of x-rays that are removed from that beam get smaller and smaller. What we can see, though, is the proportion of x-rays removed remains the same. The percentage of x-rays that we remove from the beam over set distances remains the same proportion. That's our linear attenuation coefficient, the proportion of x-rays that are removed from a beam.
If we look at this distance here, we can see that we've got 12 photons. Let's say on this y-axis, we've got 12 lines here. Initially, we've got 12 incident photons. Now, as we travel a set distance, we halve those number of photons. We have now got six photons left in that beam. If we travel that same distance, now those six photons now become three photons. So, over that first set distance, we lost six photons. Now we've lost three photons. We've lost half as many photons. However, over this distance, we have half the total number of photons. Over the next set distance, we've also halved the total number of photons, and that continues as we travel these set distances. If we go again here, we've gone from three photons to one and a half photons. And this distance is what's known as our half-value layer, the distance required in order to halve the number of x-ray photons. And you can see that half-value layer is determined by our linear attenuation coefficient here.
Now, let's have a look at this equation, and we're going to look at it in a little bit more depth when we look at our half-value layer. This equation shows us the number of x-rays that remain in our x-ray beam, the transmitted x-rays here. This N here represents the photon number left in our beam, photons that have not interacted with matter. This N naught is our incident x-ray photon number. And then we've got a negative exponent here with our linear attenuation coefficient and the distance that is traveled through the tissue. We can see that as distance increases, the number of photons that remain in our beam will decrease. This is a negative exponent here. Same goes with our linear attenuation coefficient. If our linear attenuation coefficient is higher, we've got a more dense tissue, or a higher atomic number, or an x-ray beam with lower energies, all of which will cause our linear attenuation coefficient to increase. The number of x-rays in our beam over a set distance would decrease. So, an increase in linear attenuation coefficient would make this graph much steeper, like this, and reduce the photon number quicker over smaller distances. So, a higher linear attenuation coefficient would result in a lower half-value layer. Those half-value layers would get smaller and smaller because the distance those x-rays travel would be smaller in order to decrease our photon number by half.
Now, this linear attenuation graph that we've drawn here is for a set x-ray beam traveling through a set tissue. Now, when we're taking x-rays of a patient, our tissues vary in density, and they vary in atomic number. And we also know that our x-ray beam is heterogeneous. It's not a monoenergetic beam. So, certain parts of our x-ray spectrum will have different linear attenuation coefficients to other parts. Now, when we're looking at linear attenuation coefficient, we assume a monoenergetic beam, but that doesn't account for the fact that our tissues have varying densities and varying atomic numbers.
So, let's have a look at a separate graph here. Now, these axes on our graph are different from this graph that we've just looked at here. These axes determine the linear attenuation coefficient on our y-axis, the rate of x-ray removal from that x-ray beam by our tissues. The x-axis is the photon energy, the energy of our x-ray beam. Now, we can determine a couple of things from this graph. The first thing we can see is as photon energy increases, our linear attenuation coefficient decreases in those tissues. Our ability to remove those x-rays decreases. We know that because the probability of the photoelectric effect, Compton scatter, and Rayleigh scatter all decrease with increasing x-ray energies. The second thing we can see here is that material density changes our linear attenuation coefficient. We know that fat floats on water. It's less dense than water, and we know that muscle is more dense than water, and bone is even more dense than muscle. So, as our density increases, our linear attenuation coefficient increases. The ability of these tissues to remove x-rays from a beam increases as their density increases.
Now, we can see here that if we're taking a radiograph and we want to differentiate between fat and water and muscle, lower photon energies will help us differentiate better because our linear attenuation coefficients are much different at these lower photon energies. As we increase those photon energies, our linear attenuation coefficients, especially for fat, water, and muscle, get much closer, and we get very little contrast between those tissues. And you may have seen this when we overexpose an image. Everything starts to look white. It's hard to tell the difference between the various muscle and fat planes within our radiograph, and that's because our linear attenuation coefficients have decreased and become much more similar at these higher photon energies. And this shows us that using lower photon energies will give us better contrast within our tissues.
Now, when we're looking at the linear attenuation coefficient, we are looking at x-rays traveling through a set distance of tissue, and we're looking at these factors that influence that linear attenuation coefficient. We're talking about the x-ray removal per centimeter here, per unit distance. Now, there's another coefficient that you may come across in your radiology physics studying, and that's what's known as the mass attenuation coefficient. The mass attenuation coefficient takes the material's mass into account, and the area of that tissue that is being exposed to x-rays, but accounts for changes in density.
Now, the way I like to go about thinking about this is by taking a block of water. If we were to look at this initial block here and imagine it was 100 grams of water, and then we were to freeze that water. Now, as water turns to ice, it expands. Its density gets less. It gets less dense. Ice floats on water, but the mass of that water hasn't changed. It remains 100 grams of ice versus 100 grams of water. The density has changed. The size of those 100-gram ice and water is different, but the mass is the same. Now, the mass attenuation coefficient accounts for that change in density. The mass attenuation coefficient for 100 grams of water will be the same as for 100 grams of ice. Now, our linear attenuation coefficient would have changed between the water and the ice because the density has changed. As the density has got less in our ice, ice floating on water, our linear attenuation coefficient has got less as well. Fewer x-rays will be removed over a set unit of distance within ice as opposed to water.
So, how do we go about calculating this mass attenuation coefficient? Well, it's actually quite logical. Our mass attenuation coefficient is our linear attenuation coefficient divided by the material's density. Now, we've seen in that previous example between 100 grams of water and 100 grams of ice that our linear attenuation coefficient will change between the water and the ice, and that change is proportional to the density difference between water and ice. And our mass attenuation coefficient for those 100-gram water and ice will remain the same. The photon energy hasn't changed, and the atomic number of the water and ice is still the same. It's only the density that has changed.
For me, understanding linear attenuation coefficient is much more important than mass attenuation coefficient because linear attenuation coefficient has a direct impact on the image that we take. If we were to take an x-ray of a glass of water with ice in it, we would see that ice on the x-ray. The ice wouldn't be invisible within that water, and that's because the linear attenuation coefficients are different. And that difference in linear attenuation coefficient will result in the contrast that we see in our image. So, if you're going to focus on anything here, remember that the linear attenuation coefficient is determined by three factors: the photon energy, material density, and the atomic number of that material. And the number of photons that we remove as we travel through tissue gets less and less, but that removal happens in proportion to our linear attenuation coefficient. This negative exponential removal of x-ray beams from the incident x-ray beam is what's determined by our coefficient, our linear attenuation coefficient, the rate of change of x-ray removal through the x-ray beam.
And I've mentioned briefly the half-value layer, the distance required to reduce our x-ray beam intensity by half, and that's exactly what we're going to look at in our next talk. Now, if you're studying for your Radiology Physics Part 1 exams or your registry exams, I've curated an x-ray question bank that I've linked in the top line in the description. I really do believe that will help you in getting prepared for those exams. So, if that's you, go check it out. Otherwise, I'll see you in the next talk where we're going to look closely at the half-value layer. Goodbye, everybody.